# Is there such thing as relativistic angular velocity?

## Main Question or Discussion Point

If so, wouldn't gamma would be like?

(1 - w^2 / c^2)^-1/2

or

(1 - v^2 / r^2c^2)^-1/2

## Answers and Replies

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First of all, you're better of doing a detailed calculation of it than to guess. Your expressions are not dimensionally correct.

Your first expression must be incorrect because c has units length/time, and w presumable has units 1/time since I'm guessing it is an angular velocity. So w^2/c^2 is not dimensionless as it should be since it is subtracted from the dimensionless quantity 1.

The same goes for the second expression, v^2/(r^2*c^2) is not dimensionless if v is an ordinary velocity of dimension length/time, and r is a length.

Consider a particle moving in a circle at radius r with a tangential velocity v. The gamma factor you'd get in your expressions would still be

gamma = 1/sqrt(1-v^2/c^2)

But if you want you may express it in terms of an angular velocity w by defining w := v/r and then you get

gamma = 1/sqrt(1- w^2*r^2/c^2)

Or you could use some factors of Pi if you like in your definition of the angular velocity. They would then appear in gamma aswell.

Thnking about relativistic angular velocity can lead to some interesting conclusions e.g. about the nature of electron spin.

See for example http://panda.unm.edu/Courses/Finley/P495/TermPapers/relangmom.pdf [Broken]

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First of all, you're better of doing a detailed calculation of it than to guess. Your expressions are not dimensionally correct.

Your first expression must be incorrect because c has units length/time, and w presumable has units 1/time since I'm guessing it is an angular velocity. So w^2/c^2 is not dimensionless as it should be since it is subtracted from the dimensionless quantity 1.

The same goes for the second expression, v^2/(r^2*c^2) is not dimensionless if v is an ordinary velocity of dimension length/time, and r is a length.

Consider a particle moving in a circle at radius r with a tangential velocity v. The gamma factor you'd get in your expressions would still be

gamma = 1/sqrt(1-v^2/c^2)

But if you want you may express it in terms of an angular velocity w by defining w := v/r and then you get

gamma = 1/sqrt(1- w^2*r^2/c^2)

Or you could use some factors of Pi if you like in your definition of the angular velocity. They would then appear in gamma aswell.
So that means
gamma = 1/sqrt(1- w^2*r^2/c^2)

is the correct formula to use. Also, nobody answered that there really is relativistic angular velocity. Does something way more if it spins really really fast?

So that means
gamma = 1/sqrt(1- w^2*r^2/c^2)

is the correct formula to use. Also, nobody answered that there really is relativistic angular velocity. Does something way more if it spins really really fast?
I think your formula is correct. As for the question, 'does something weigh more if it spins fast', yes it would seem so. Think about angular momentum, not angular velocity ... it should involve the factor, gamma, somehow, though it may not be a simple formula. One must integrate over r to find total angular momentum, as a function of the object's angular velocity.